Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces

Best approximation in Chebyshev subspaces of L(l1n,l1n)

“…This result is quite different from the result obtained for the space L(l n 1 , c 0 ) (see [6]), where any finite dimensional Chebyshev subspace is an interpolating subspace. Additionally, as the space L(l n 1 , l n 1 ) is a finite dimensional space, we get (see [13]) that the unicity of best approximation is equivalent to the strong unicity of best approximation.…”

Best Approximation and Best Simultaneous Approximation in Ultrametric Spaces

“…It is well known that a compact set in a metric space is proximinal. The condition of compactness has been weakened to approximative compactness (Efimov and Steckin, see [9]), bounded compactness (V. Klee, see [6]), and spherical compactness (G. Albinus, see [6]). Here in this section we introduce analogous notions in ultrametric spaces and prove certain existence theorems on best approximation.…”

“…In archimedean theory, it was proved by Efimov and Steckin (see [9]) that an approximatively compact subset of a metric space is proximinal. In non-archimedean theory we have the following result By the strong triangle inequality, we then have…”

“…[1], [5] and [9]). In 1956, A. F. Monna [3] initiated the study of best approximation in non-archimedean normed linear spaces over valued fields other than R or C. But not much is known about best approximation in ultrametric spaces, i.e.…”

Best Approximation in Metric Spaces

“…[2], [3], [12], [13] and references cited therein) but only a few have taken up this study in more general abstract spaces viz. metric linear spaces, convex metric spaces and metric spaces (see e.g.…”